Showing papers in "Journal of Elasticity in 2006"
TL;DR: In this article, the closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry, where the mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity Tensor of the specified symmetry.
Abstract: The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.
204 citations
TL;DR: In this paper, a theoretical framework for the analysis of boundary-value problems that underpin the applications of the associated electromechanical interactions is presented, where the authors consider the influence of a radial electric field on the azimuthal shear response of a thick-walled circular cylindrical tube, the extension and inflation characteristics of the same tube under either a radial or an axial electric field (or both fields combined), and the effect of an internally pressurized spherical shell.
Abstract: Electro-sensitive (ES) elastomers form a class of smart materials whose mechanical properties can be changed rapidly by the application of an electric field. These materials have attracted considerable interest recently because of their potential for providing relatively cheap and light replacements for mechanical devices, such as actuators, and also for the development of artificial muscles. In this paper we are concerned with a theoretical framework for the analysis of boundary-value problems that underpin the applications of the associated electromechanical interactions. We confine attention to the static situation and first summarize the governing equations for a solid material capable of large electroelastic deformations. The general constitutive laws for the Cauchy stress tensor and the electric field vectors for an isotropic electroelastic material are developed in a compact form following recent work by the authors. The equations are then applied, in the case of an incompressible material, to the solution of a number of representative boundary-value problems. Specifically, we consider the influence of a radial electric field on the azimuthal shear response of a thick-walled circular cylindrical tube, the extension and inflation characteristics of the same tube under either a radial or an axial electric field (or both fields combined), and the effect of a radial field on the deformation of an internally pressurized spherical shell.
196 citations
TL;DR: In this article, the viscoelastic Poisson ratio has a different time dependence depending on the test modality chosen; interrelations are developed between Poisson's ratios in creep and relaxation.
Abstract: Poisson’s ratio in viscoelastic solids is in general a time dependent (in the time domain) or a complex frequency dependent quantity (in the frequency domain) We show that the viscoelastic Poisson’s ratio has a different time dependence depending on the test modality chosen; interrelations are developed between Poisson’s ratios in creep and relaxation The difference, for a moderate degree of viscoelasticity, is minor Correspondence principles are derived for the Poisson’s ratio in transient and dynamic contexts The viscoelastic Poisson’s ratio need not increase with time, and it need not be monotonic with time Examples are given of material microstructures which give rise to designed time dependent Poisson’s ratios
182 citations
TL;DR: In this article, the authors present properly invariant averaging procedures for symmetric positive-definite tensors which are based on different measures of nearness of symmetric PDF tensors.
Abstract: In this paper we present properly invariant averaging procedures for symmetric positive-definite tensors which are based on different measures of nearness of symmetric positive-definite tensors. These procedures intrinsically account for the positive-definite property of the tensors to be averaged. They are independent of the coordinate system, preserve material symmetries, and more importantly, they are invariant under inversion. The results of these averaging methods are compared with the results of other methods including that proposed by Cowin and Yang (J. of Elasticity 46 (1997) pp. 151–180.) for the case of the elasticity tensor of generalized Hooke's law.
172 citations
TL;DR: In this article, a quantitative study of the size of representative volume element (RVE) of random matrix-inclusion composites based on a scale-dependent homogenization method is presented.
Abstract: This paper presents a quantitative study of the size of representative volume element (RVE) of random matrix-inclusion composites based on a scale-dependent homogenization method. In particular, mesoscale bounds defined under essential or natural boundary conditions are computed for several nonlinear elastic, planar composites, in which the matrix and inclusions differ not only in their material parameters but also in their strain energy function representations. Various combinations of matrix and inclusion phases described by either neo-Hookean or Ogden function are examined, and these are compared to those of linear elastic types.
128 citations
TL;DR: The local symmetry group of the dynamically and kinematically exact theory of elastic shells was established in this article, which consists of an ordered triple of tensors which make the shell strain energy density invariant under change of the reference placement.
Abstract: We establish the local symmetry group of the dynamically and kinematically exact theory of elastic shells. The group consists of an ordered triple of tensors which make the shell strain energy density invariant under change of the reference placement. Definitions of the fluid shell, the solid shell, and the membrane shell are introduced in terms of members of the symmetry group. Within solid shells we discuss in more detail the isotropic, hemitropic, and orthotropic shells and corresponding invariant properties of the strain energy density. For the physically linear shells, when the density becomes a quadratic function of the shell strain and bending tensors, reduced representations of the density are established for orthotropic, cubic-symmetric, and isotropic shells. The reduced representations contain much less independent material constants to be found from experiments.
112 citations
TL;DR: In this article, a discrete homogenization technique was used to construct an equivalent macroscopic model for large three-dimensional deformations of graphene sheets, which is valid for small displacements and whose constitutive laws are explicit.
Abstract: Graphene sheets can be considered as lattices consisting of atoms and of interatomic bonds. Their bond lengths are smaller than one nanometer. Simple models describe their behavior by an energy that takes into account both the interatomic lengths and the angles between bonds. We make use of their periodic structure and we construct an equivalent macroscopic model by means of a discrete homogenization technique. Large three-dimensional deformations of graphene sheets are thus governed by a membrane model whose constitutive law is implicit. By linearizing around a prestressed configuration, we obtain linear membrane models that are valid for small displacements and whose constitutive laws are explicit. When restricting to two-dimensional deformations, we can linearize around a rest configuration and we provide explicit macroscopical mechanical constants expressed in terms of the interatomic tension and bending stiffnesses.
85 citations
TL;DR: In this article, the existence of solutions to hemivariational inequalities with different types of non-monotone multivalued boundary relations is established for both unilateral and bilateral contact mechanics.
Abstract: In this paper we consider mathematical models describing dynamic viscoelastic contact problems with the Kelvin–Voigt constitutive law and subdifferential boundary conditions. We treat evolution hemivariational inequalities which are weak formulations of contact problems. We establish the existence of solutions to hemivariational inequalities with different types of non-monotone multivalued boundary relations. These results are consequences of an existence theorem for second order evolution inclusions. In a particular case we deliver sufficient conditions under which the solution to a hemivariational inequality is unique. Finally, applications to several unilateral and bilateral problems in contact mechanics are provided.
82 citations
TL;DR: Benham and Le Bret as discussed by the authors showed that the problem had been completely solved by John Henry Michell in 1889 in a rather elegant manner and to reflect on its history and modern developments.
Abstract: Elastic rings become unstable when sufficiently twisted. This fundamental instability plays an important role in the modeling of DNA mechanics and in cable engineering. In 1962, Zajac computed the value of the critical twist for the instability. This critical value was rediscovered in 1979 by Benham and independently by Le Bret in elastic models for DNA; unstable rings have since become an important example of elastic instabilities in rods both for the development of new methods and in applications. The purpose of this note is to show that the problem had been completely solved by John Henry Michell in 1889 in a rather elegant manner and to reflect on its history and modern developments.
75 citations
TL;DR: A theoretical method of calculating the speed at which the ventricular walls move inwardly is reported on, which will become of great practical usefulness as soon as more experimental results become available.
Abstract: Hydrocephalus is a condition which occurs when an excessive accumulation of cerebrospinal fluid in the brain causes enlargement of the ventricular cavities. Modern treatments of shunt implantation are effective, but have an unacceptably high rate of failure in most reported series. One of the common factors causing shunt failure is the misplacement of the proximal catheter's tip, which can be remedied if the healed configuration of the ventricular space can be predicted. In a recent study we have shown that this is accomplished by a mathematical model which requires as input the knowledge of the speed at which the ventricular walls move inwardly. In this paper we report on a theoretical method of calculating this speed and show that it will become of great practical usefulness as soon as more experimental results become available.
52 citations
TL;DR: In this paper, the dual conservation laws of elasticity are systematically re-examined by using both Noether's variational approach and Coleman-Noll-Gurtin's thermodynamics approach.
Abstract: The dual conservation laws of elasticity are systematically re-examined by using both Noether's variational approach and Coleman-Noll-Gurtin's thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy-momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy-momentum tensor) P and the dual configuration force (dual energy-momentum tensor) L, PL ¼ð P : FÞ1 �r ð PxÞ: This and other results derived in this paper may lead to a better understanding of configurational mechanics and therefore of mechanics of defects.
TL;DR: In this article, the authors derived and mathematically justified models of micropolar rods and plates from the equations of linearized micropolastic elasticity based on the asymptotic techniques with respect to the small parameter being the thickness of the elastic body.
Abstract: In this paper we derive and mathematically justify models of micropolar rods and plates from the equations of linearized micropolar elasticity. Derivation is based on the asymptotic techniques with respect to the small parameter being the thickness of the elastic body we consider. Justification of the models is obtained through the convergence result for the displacement and microrotation fields when the thickness tends to zero. The limiting microrotation is then related to the macrorotation of the cross–section (transversal segment) and the model is rewritten in terms of macroscopic unknowns. The obtained models are recognized as being either the Reissner–Mindlin plate or the Timoshenko beam type.
TL;DR: In this article, it was shown that for anisotropic elastic materials, only certain orthotropic and hexagonal materials can have Young modulus independent of the direction n or the shear modulus G(n,m) independent of n and m. The structure of elastic compliance for these materials is presented, and condition for identifying if the material is orthotropic or hexagonal is given.
Abstract: It is shown that, among anisotropic elastic materials, only certain orthotropic and hexagonal materials can have Young modulus E(n) independent of the direction n or the shear modulus G(n,m) independent of n and m. Thus the direction surface for E(n) can be a sphere for certain orthotropic and hexagonal materials. The structure of the elastic compliance for these materials is presented, and condition for identifying if the material is orthotropic or hexagonal is given. We also study the case in which n of E(n) and n, m of G(n,m) are restricted to a plane. When E(n) is a constant on a plane so are G(n,m) and Poisson's ratio ν(n,m). The converse, however, does not necessarily hold. A plane on which E(n) is a constant can exist for all anisotropic elastic materials. In particular, existence of such a plane is assured for trigonal, hexagonal and cubic materials. In fact there are four such planes for a cubic material. For these materials, not only E(n) is a constant, two other Young's moduli, the three shear moduli and the six Poisson's ratio on the plane are also constant.
TL;DR: In this article, the authors make the principle of material frame-indifference vacuously satisfied by using an intrinsic mathematical frame-work that does not use an external frame-space at all when describing the internal interactions of a physical system.
Abstract: As I pointed out at the end of Sect. 4 in [6] of my booklet Five Contributions to Natural Philosophy, it should be possible to make the principle of material frame-indifference vacuously satisfied by using an intrinsic mathematical frame-work that does not use an external frame-space at all when describing the internal interactions of a physical system. Here I will do just that for the classical theory of elasticity and also for the theory of hyperelasticity, i.e., elasticity based on a strain-energy function. I will also comment on possible restrictions on the corresponding intrinsic response functions.
TL;DR: In this article, the authors use formal asymptotic expansions to build one-dimensional models for slender hyperelastic cylinders submitted to conservative loads, and obtain a hierarchy of models going from the linear theory of flexible bars to the nonlinear theory of extensible strings.
Abstract: By using formal asymptotic expansions, we build one-dimensional models for slender hyperelastic cylinders submitted to conservative loads. According to the order of magnitude of the applied loads, we obtain a hierarchy of models going from the linear theory of flexible bars to the nonlinear theory of extensible strings.
TL;DR: In this paper, a theory of anharmonic lattice statics for the analysis of defective complex lattices is developed, which does not rely on harmonic and homogenous force constants.
Abstract: This paper develops a theory of anharmonic lattice statics for the analysis of defective complex lattices. This theory differs from the classical treatments of defects in lattice statics in that it does not rely on harmonic and homogenous force constants. Instead, it starts with an interatomic potential, possibly with infinite range as appropriate for situations with electrostatics, and calculates the equilibrium states of defects. In particular, the present theory accounts for the differences in the force constants near defects and in the bulk. The present formulation reduces the analysis of defective crystals to the solution of a system of nonlinear difference equations with appropriate boundary conditions. A harmonic problem is obtained by linearizing the nonlinear equations, and a method for obtaining analytical solutions is described in situations where one can exploit symmetry. It is then extended to the anharmonic problem using modified Newton–Raphson iteration. The method is demonstrated for model problems motivated by domain walls in ferroelectric materials.
TL;DR: In this paper, the inverse problem of identifying the density and elastic moduli for three-dimensional anisotropic elastic bodies, given displacement and traction measurements made at their surface, is considered.
Abstract: We consider the inverse problem of identifying the density and elastic moduli for three-dimensional anisotropic elastic bodies, given displacement and traction measurements made at their surface. These surface measurements are modelled by the dynamic Dirichlet-to-Neumann map on a finite time interval. For linear or nonlinear anisotropic hyperelastic bodies we show that the displacement-to-traction surface measurements do not change when the density and elasticity tensor in the interior are transformed tensorially by a change of coordinates fixing the surface of the body to first order. Our main tool, a new approach in inverse problems for elastic media, is the representation of the equations of motion in a covariant form (following Marsden and Hughes, 1983) that preserves the underlying physics.
TL;DR: In this paper, it is shown that locally homogeneous deformations can be produced in such inhomogeneous materials and form a differentiable manifold, and connections are also established between the effective nominal and material elastic tangent moduli.
Abstract: This work is concerned with the homogenization of solids reinforced by aligned parallel continuous fibers or weakened by aligned parallel cylindrical pores and undergoing large deformations. By alternatively exploiting the nominal and material formulations of the corresponding homogenization problem and by applying the implicit function theorem, it is shown that locally homogeneous deformations can be produced in such inhomogeneous materials and form a differentiable manifold. For every macroscopic strain associated to a locally homogeneous deformation field, the effective nominal or material stress-strain relation is exactly determined and connections are also exactly established between the effective nominal and material elastic tangent moduli. These results are microstructure-independent in the sense that they hold irrespectively of the transverse geometry and distribution of the fibers or pores. A porous medium consisting of a compressible Mooney-Rivlin material with cylindrical pores is studied in detail to illustrate the general results.
TL;DR: In this article, it was shown that the displacement and stress fields in the inhomogeneous material can be expressed in terms of four complex potentials that are analytic functions of the complex variable ζ = x + iy in the midplane of the plate.
Abstract: A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a thick plate of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical thin plate or classical laminate theory equations (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the homogeneous material. Recently this theory has been formulated in terms of functions of a complex variable. It was shown that the displacement and stress fields in the inhomogeneous material could be expressed in terms of four complex potentials that are analytic functions of the complex variable ζ = x + iy in the mid-plane of the plate. However, the analysis performed so far applies only to the case of a plate with traction-free upper and lower faces. The present paper extends these solutions to the case where the plate is bent by a pressure distribution applied to a face.
TL;DR: In this paper, the generalized radiation condition for time-harmonic elastodynamic states in a piecewise-homogeneous, semi-infinite solid was established, where the bottom homogeneous half-space is overlain by an arbitrary number of bonded parallel layers.
Abstract: This study aims to establish a generalized radiation condition for time-harmonic elastodynamic states in a piecewise-homogeneous, semi-infinite solid wherein the “bottom” homogeneous half-space is overlain by an arbitrary number of bonded parallel layers. To consistently deal with both body and interfacial (e.g. Rayleigh, Love and Stoneley) waves comprising the far-field patterns, the radiation condition is formulated in terms of an integral over a sufficiently large hemisphere involving elastodynamic Green's functions for the featured layered medium. On explicitly proving the reciprocity identity for the latter set of point-load solutions, it is first shown that the layered Green's functions themselves satisfy the generalized radiation condition. By virtue of this result it is further demonstrated that the entire class of layered elastodynamic solutions, admitting a representation in terms of the single-layer, double-layer, and volume potentials (distributed over finite domains), satisfy the generalized radiation condition as well. For a rigorous treatment of the problem, fundamental results such as the uniqueness theorem for radiating elastodynamic states, Graffi's reciprocity theorem for piecewise-homogeneous domains, and the integral representation theorem for semi-infinite layered media are also established.
TL;DR: In this paper, the authors derived a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value in a macroscopically homogeneous anisotropic, prestressed half-space.
Abstract: Herein we consider Rayleigh waves propagating along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part \(\mathbb{A}\) of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if \(\mathbb{A}\) were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.
TL;DR: In this article, a new proof of the representation theorem for fourth-order isotropic tensors was presented that does not assume the tensor to have major or minor symmetries at the outset.
Abstract: We present a new proof of the representation theorem for fourth-order isotropic tensors that does not assume the tensor to have major or minor symmetries at the outset.
TL;DR: In this paper, the Fourier transform method is applied to the Hertzian contact problem for anisotropic piezoelectric bodies, and the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations.
Abstract: The Fourier transform method is applied to the Hertzian contact problem for anisotropic piezoelectric bodies. Using the principle of linear superposition, the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations, one containing pressure and another containing electric displacement. By presupposing the forms of the pressure and electric displacement distribution over the contact area, the problem is solved successfully; then the generalized displacements, stresses and strains are expressed by contour integrals. Details are presented in the case of special orthotropic piezoelectricity whose material constants satisfy six relations, which can be easily degenerated to the case of transverse isotropic piezoelectricity. It can be shown that the result gained in this paper is of a universal and compact form for a general material.
TL;DR: In this article, the authors apply the asymptotic analysis procedure to the three-dimensional static equations of piezoelectricity, for a linear nonhomogeneous anisotropic thin rod.
Abstract: We apply the asymptotic analysis procedure to the three-dimensional static equations of piezoelectricity, for a linear nonhomogeneous anisotropic thin rod. We prove the weak convergence of the rod mechanical displacement vectors and the rod electric potentials, when the diameter of the rod cross-section tends to zero. This weak limit is the solution of a new piezoelectric anisotropic nonhomogeneous rod model, which is a system of coupled equations, with generalized Bernoulli–Navier equilibrium equations and reduced Maxwell–Gauss equations.
TL;DR: In this article, it was shown that any stress response function which is assumed to depend upon the motion in an inertial (general) frame must be Galilean-invariant under superposed rigid body motions.
Abstract: In Continuum Mechanics the notions of body, material point, and motion, are primitive. Here these concepts are derived for any (possibly time-dependent) material system via mass and momentum densities whose values are local spacetime averages of molecular quantities. The averaging procedure necessary to ensure molecular-based densities can be agreed upon by all observers (that is, are objective) has implications for constitutive relations. Specifically, such relations should first be expressed in terms of Galilean-invariant functions of the motion relative to an inertial frame. Thereafter such relations can be re-phrased for general observers, thereby yielding general-frame constitutive relations compatible with material frame-indifference. Two postulates concerning observer agreement (which together constitute a statement of material frame-indifference) are shown to imply that any stress response function which is assumed to depend upon the motion in an inertial (general) frame must be Galilean-invariant (invariant under superposed rigid body motions). Accordingly, invariance under superposed rigid body motions is not a fundamental tenet of continuum physics, but rather a consequence of material frame-indifference whenever constitutive dependence upon motion in a general observer frame is postulated.
TL;DR: In this article, the influence of elastic inhomogeneity on the elastic field in solid circular cylinders, pipes, and also in a solid of infinite extent surrounding an internally pressurized cavity is investigated.
Abstract: The purpose of this paper is to investigate the influence of elastic inhomogeneity on the elastic field in solid circular cylinders, pipes, and also in a solid of infinite extent surrounding an internally pressurized cavity. The motivation for this research stems from recent interest in the use of laser technology and the vapor deposition of thin layers onto the surfaces of pipes. The present analysis extends previous treatments insofar as a more general form for the spatial variation of Young's modulus is used. An estimate for the optimal functional gradient within a pressurized pipe is presented.
TL;DR: A solution for the tensor equation TX + XTT = H, where T is a diagonalizable tensor, which is valid for any arbitrary underlying vector space dimension n, is presented and compact expressions for the derivatives of the stretch and rotation tensors are derived.
Abstract: We present a solution for the tensor equation TX + XTT =H, where T is a diagonalizable ( in particular, symmetric tensor, which is valid for any arbitrary underlying vector space dimension n. This solution is then used to derive compact expressions for the derivatives of the stretch and rotation tensors, which in turn are used to derive expressions for the material time derivatives of these tensors. Some existing expressions for n = 2 and n = 3 are shown to follow from the presented solution as special cases. An alternative methodology for finding the derivatives of diagonalizable tensor-valued functions that is based on differentiating the spectral decomposition is also discussed. Lastly, we also present a method for finding the derivatives of the exponential of an arbitrary tensor for arbitrary n.
TL;DR: In this article, the boundary value problem of plane micropolar elasticity for a domain containing a crack in Sobolev spaces was formulated and the existence and continuous dependence on the data of the corresponding weak solutions were proved.
Abstract: In this paper we formulate the boundary value problem of plane micropolar elasticity for a domain containing a crack in Sobolev spaces and prove the existence and continuous dependence on the data of the corresponding weak solutions. We consider the cases of both finite and infinite domain and find the solutions in terms of modified single layer and modified double layer potentials with distributional densities.
TL;DR: In this paper, a general framework for handling a broad range of non-uniform distribution of eigenstrains exactly, but also has great implications in employing the equivalent inclusion method (EIM) to study the behavior of composites with functionally graded reinforcements.
Abstract: In association with multi-inhomogeneity problems, a special class of eigenstrains is discovered to give rise to disturbance stresses of interesting nature. Some previously unnoticed properties of Eshelby’s tensors prove useful in this accomplishment. Consider the set of nested similar ellipsoidal domains {Ω1, Ω2,⋯,ΩN+1}, which are embedded in an infinite isotropic medium. Suppose that
$$\Omega_{t}=\{\; {\textbf{x}} \mid {\textbf{x}} \in \mathbb{R}^{3} , \quad \sum_{p=1}^3\frac{x_{p}^2}{a_{p}^{2}}\leqslant\xi_{t}^2 \;\},$$
in which \(0\leqslant\xi_{1} < \xi_{2} < \cdots < \xi_{N+1}\) and ξtap, p=1,2,3 are the principal half axes of Ωt. Suppose, the distribution of eigenstrain, eij*(x) over the regions Γt=Ωt+1−Ωt , t=1,2,⋯,N can be expressed as
$$\epsilon_{ij}^{*} \left( {\textbf{x}} \right)=\begin{cases} f_{ijkl \cdots m }^{(t)}\biggl( \sum\limits_{p=1}^3\dfrac{ x_{p}^2 }{a_{p}^{2}} \biggr) \;x_{k}x_{l} \cdots x_{m}, \qquad {\textbf{x}} \in \Gamma_{t},\\ \quad 0, \qquad\qquad\qquad\qquad\qquad\ \qquad {\textbf{x}} \in \Omega_1\bigcup \left(\mathbb{R}^3-\Omega_{N+1}\right), \end{cases}$$
(‡)
where xkxl ⋯xm is of order n, and fijkl ⋯m(t) represents 3N(n+2)(n+1) different piecewise continuous functions whose arguments are ∑p=13xp2 /ap2. The nature of the disturbance stresses due to various classes of the piecewise nonuniform distribution of eigenstrains, obtained via superpositions of Eq. (‡) is predicted and an infinite number of impotent eigenstrains are introduced. The present theory not only provides a general framework for handling a broad range of nonuniform distribution of eigenstrains exactly, but also has great implications in employing the equivalent inclusion method (EIM) to study the behavior of composites with functionally graded reinforcements.
TL;DR: In this article, the authors derived compact basis-free representations for the time rate and conjugate stress of logarithmic strain tensors using six different methods and validated them for the cases of distinct stretches and double coalescence.
Abstract: In this paper, two kinds of tensor equations are studied and their solutions are derived in general cases. Then, some compact basis-free representations for the time rate and conjugate stress of logarithmic strain tensors are proposed using six different methods. In addition, relations between the coefficients in these expressions are disclosed. Subsequently, all these basis-free expressions given in this paper are validated for the cases of distinct stretches and double coalescence, respectively.